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LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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| subroutine dchkst2stg | ( | integer | NSIZES, |
| integer, dimension( * ) | NN, | ||
| integer | NTYPES, | ||
| logical, dimension( * ) | DOTYPE, | ||
| integer, dimension( 4 ) | ISEED, | ||
| double precision | THRESH, | ||
| integer | NOUNIT, | ||
| double precision, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| double precision, dimension( * ) | AP, | ||
| double precision, dimension( * ) | SD, | ||
| double precision, dimension( * ) | SE, | ||
| double precision, dimension( * ) | D1, | ||
| double precision, dimension( * ) | D2, | ||
| double precision, dimension( * ) | D3, | ||
| double precision, dimension( * ) | D4, | ||
| double precision, dimension( * ) | D5, | ||
| double precision, dimension( * ) | WA1, | ||
| double precision, dimension( * ) | WA2, | ||
| double precision, dimension( * ) | WA3, | ||
| double precision, dimension( * ) | WR, | ||
| double precision, dimension( ldu, * ) | U, | ||
| integer | LDU, | ||
| double precision, dimension( ldu, * ) | V, | ||
| double precision, dimension( * ) | VP, | ||
| double precision, dimension( * ) | TAU, | ||
| double precision, dimension( ldu, * ) | Z, | ||
| double precision, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | LIWORK, | ||
| double precision, dimension( * ) | RESULT, | ||
| integer | INFO | ||
| ) |
DCHKST2STG
DCHKST2STG checks the symmetric eigenvalue problem routines
using the 2-stage reduction techniques. Since the generation
of Q or the vectors is not available in this release, we only
compare the eigenvalue resulting when using the 2-stage to the
one considered as reference using the standard 1-stage reduction
DSYTRD. For that, we call the standard DSYTRD and compute D1 using
DSTEQR, then we call the 2-stage DSYTRD_2STAGE with Upper and Lower
and we compute D2 and D3 using DSTEQR and then we replaced tests
3 and 4 by tests 11 and 12. test 1 and 2 remain to verify that
the 1-stage results are OK and can be trusted.
This testing routine will converge to the DCHKST in the next
release when vectors and generation of Q will be implemented.
DSYTRD factors A as U S U' , where ' means transpose,
S is symmetric tridiagonal, and U is orthogonal.
DSYTRD can use either just the lower or just the upper triangle
of A; DCHKST2STG checks both cases.
U is represented as a product of Householder
transformations, whose vectors are stored in the first
n-1 columns of V, and whose scale factors are in TAU.
DSPTRD does the same as DSYTRD, except that A and V are stored
in "packed" format.
DORGTR constructs the matrix U from the contents of V and TAU.
DOPGTR constructs the matrix U from the contents of VP and TAU.
DSTEQR factors S as Z D1 Z' , where Z is the orthogonal
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal. D2 is the matrix of
eigenvalues computed when Z is not computed.
DSTERF computes D3, the matrix of eigenvalues, by the
PWK method, which does not yield eigenvectors.
DPTEQR factors S as Z4 D4 Z4' , for a
symmetric positive definite tridiagonal matrix.
D5 is the matrix of eigenvalues computed when Z is not
computed.
DSTEBZ computes selected eigenvalues. WA1, WA2, and
WA3 will denote eigenvalues computed to high
absolute accuracy, with different range options.
WR will denote eigenvalues computed to high relative
accuracy.
DSTEIN computes Y, the eigenvectors of S, given the
eigenvalues.
DSTEDC factors S as Z D1 Z' , where Z is the orthogonal
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). It may also
update an input orthogonal matrix, usually the output
from DSYTRD/DORGTR or DSPTRD/DOPGTR ('V' option). It may
also just compute eigenvalues ('N' option).
DSTEMR factors S as Z D1 Z' , where Z is the orthogonal
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). DSTEMR
uses the Relatively Robust Representation whenever possible.
When DCHKST2STG is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the symmetric eigenroutines. For each matrix, a number
of tests will be performed:
(1) | A - V S V' | / ( |A| n ulp ) DSYTRD( UPLO='U', ... )
(2) | I - UV' | / ( n ulp ) DORGTR( UPLO='U', ... )
(3) | A - V S V' | / ( |A| n ulp ) DSYTRD( UPLO='L', ... )
replaced by | D1 - D2 | / ( |D1| ulp ) where D1 is the
eigenvalue matrix computed using S and D2 is the
eigenvalue matrix computed using S_2stage the output of
DSYTRD_2STAGE("N", "U",....). D1 and D2 are computed
via DSTEQR('N',...)
(4) | I - UV' | / ( n ulp ) DORGTR( UPLO='L', ... )
replaced by | D1 - D3 | / ( |D1| ulp ) where D1 is the
eigenvalue matrix computed using S and D3 is the
eigenvalue matrix computed using S_2stage the output of
DSYTRD_2STAGE("N", "L",....). D1 and D3 are computed
via DSTEQR('N',...)
(5-8) Same as 1-4, but for DSPTRD and DOPGTR.
(9) | S - Z D Z' | / ( |S| n ulp ) DSTEQR('V',...)
(10) | I - ZZ' | / ( n ulp ) DSTEQR('V',...)
(11) | D1 - D2 | / ( |D1| ulp ) DSTEQR('N',...)
(12) | D1 - D3 | / ( |D1| ulp ) DSTERF
(13) 0 if the true eigenvalues (computed by sturm count)
of S are within THRESH of
those in D1. 2*THRESH if they are not. (Tested using
DSTECH)
For S positive definite,
(14) | S - Z4 D4 Z4' | / ( |S| n ulp ) DPTEQR('V',...)
(15) | I - Z4 Z4' | / ( n ulp ) DPTEQR('V',...)
(16) | D4 - D5 | / ( 100 |D4| ulp ) DPTEQR('N',...)
When S is also diagonally dominant by the factor gamma < 1,
(17) max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
DSTEBZ( 'A', 'E', ...)
(18) | WA1 - D3 | / ( |D3| ulp ) DSTEBZ( 'A', 'E', ...)
(19) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
DSTEBZ( 'I', 'E', ...)
(20) | S - Y WA1 Y' | / ( |S| n ulp ) DSTEBZ, SSTEIN
(21) | I - Y Y' | / ( n ulp ) DSTEBZ, SSTEIN
(22) | S - Z D Z' | / ( |S| n ulp ) DSTEDC('I')
(23) | I - ZZ' | / ( n ulp ) DSTEDC('I')
(24) | S - Z D Z' | / ( |S| n ulp ) DSTEDC('V')
(25) | I - ZZ' | / ( n ulp ) DSTEDC('V')
(26) | D1 - D2 | / ( |D1| ulp ) DSTEDC('V') and
DSTEDC('N')
Test 27 is disabled at the moment because DSTEMR does not
guarantee high relatvie accuracy.
(27) max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
DSTEMR('V', 'A')
(28) max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
DSTEMR('V', 'I')
Tests 29 through 34 are disable at present because DSTEMR
does not handle partial spectrum requests.
(29) | S - Z D Z' | / ( |S| n ulp ) DSTEMR('V', 'I')
(30) | I - ZZ' | / ( n ulp ) DSTEMR('V', 'I')
(31) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
DSTEMR('N', 'I') vs. SSTEMR('V', 'I')
(32) | S - Z D Z' | / ( |S| n ulp ) DSTEMR('V', 'V')
(33) | I - ZZ' | / ( n ulp ) DSTEMR('V', 'V')
(34) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
DSTEMR('N', 'V') vs. SSTEMR('V', 'V')
(35) | S - Z D Z' | / ( |S| n ulp ) DSTEMR('V', 'A')
(36) | I - ZZ' | / ( n ulp ) DSTEMR('V', 'A')
(37) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
DSTEMR('N', 'A') vs. SSTEMR('V', 'A')
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (4), but multiplied by SQRT( overflow threshold )
(7) Same as (4), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U' D U, where U is orthogonal and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U' D U, where U is orthogonal and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U' D U, where U is orthogonal and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Symmetric matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
(16) Same as (8), but diagonal elements are all positive.
(17) Same as (9), but diagonal elements are all positive.
(18) Same as (10), but diagonal elements are all positive.
(19) Same as (16), but multiplied by SQRT( overflow threshold )
(20) Same as (16), but multiplied by SQRT( underflow threshold )
(21) A diagonally dominant tridiagonal matrix with geometrically
spaced diagonal entries 1, ..., ULP. | [in] | NSIZES | NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
DCHKST2STG does nothing. It must be at least zero. |
| [in] | NN | NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero. |
| [in] | NTYPES | NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, DCHKST2STG
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. . |
| [in] | DOTYPE | DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored. |
| [in,out] | ISEED | ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to DCHKST2STG to continue the same random number
sequence. |
| [in] | THRESH | THRESH is DOUBLE PRECISION
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero. |
| [in] | NOUNIT | NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.) |
| [in,out] | A | A is DOUBLE PRECISION array of
dimension ( LDA , max(NN) )
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually
used. |
| [in] | LDA | LDA is INTEGER
The leading dimension of A. It must be at
least 1 and at least max( NN ). |
| [out] | AP | AP is DOUBLE PRECISION array of
dimension( max(NN)*max(NN+1)/2 )
The matrix A stored in packed format. |
| [out] | SD | SD is DOUBLE PRECISION array of
dimension( max(NN) )
The diagonal of the tridiagonal matrix computed by DSYTRD.
On exit, SD and SE contain the tridiagonal form of the
matrix in A. |
| [out] | SE | SE is DOUBLE PRECISION array of
dimension( max(NN) )
The off-diagonal of the tridiagonal matrix computed by
DSYTRD. On exit, SD and SE contain the tridiagonal form of
the matrix in A. |
| [out] | D1 | D1 is DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by DSTEQR simlutaneously
with Z. On exit, the eigenvalues in D1 correspond with the
matrix in A. |
| [out] | D2 | D2 is DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by DSTEQR if Z is not
computed. On exit, the eigenvalues in D2 correspond with
the matrix in A. |
| [out] | D3 | D3 is DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by DSTERF. On exit, the
eigenvalues in D3 correspond with the matrix in A. |
| [out] | D4 | D4 is DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by DPTEQR(V).
DPTEQR factors S as Z4 D4 Z4*
On exit, the eigenvalues in D4 correspond with the matrix in A. |
| [out] | D5 | D5 is DOUBLE PRECISION array of
dimension( max(NN) )
The eigenvalues of A, as computed by DPTEQR(N)
when Z is not computed. On exit, the
eigenvalues in D4 correspond with the matrix in A. |
| [out] | WA1 | WA1 is DOUBLE PRECISION array of
dimension( max(NN) )
All eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by DSTEBZ. |
| [out] | WA2 | WA2 is DOUBLE PRECISION array of
dimension( max(NN) )
Selected eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by DSTEBZ.
Choose random values for IL and IU, and ask for the
IL-th through IU-th eigenvalues. |
| [out] | WA3 | WA3 is DOUBLE PRECISION array of
dimension( max(NN) )
Selected eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by DSTEBZ.
Determine the values VL and VU of the IL-th and IU-th
eigenvalues and ask for all eigenvalues in this range. |
| [out] | WR | WR is DOUBLE PRECISION array of
dimension( max(NN) )
All eigenvalues of A, computed to high
absolute accuracy, with different options.
as computed by DSTEBZ. |
| [out] | U | U is DOUBLE PRECISION array of
dimension( LDU, max(NN) ).
The orthogonal matrix computed by DSYTRD + DORGTR. |
| [in] | LDU | LDU is INTEGER
The leading dimension of U, Z, and V. It must be at least 1
and at least max( NN ). |
| [out] | V | V is DOUBLE PRECISION array of
dimension( LDU, max(NN) ).
The Housholder vectors computed by DSYTRD in reducing A to
tridiagonal form. The vectors computed with UPLO='U' are
in the upper triangle, and the vectors computed with UPLO='L'
are in the lower triangle. (As described in DSYTRD, the
sub- and superdiagonal are not set to 1, although the
true Householder vector has a 1 in that position. The
routines that use V, such as DORGTR, set those entries to
1 before using them, and then restore them later.) |
| [out] | VP | VP is DOUBLE PRECISION array of
dimension( max(NN)*max(NN+1)/2 )
The matrix V stored in packed format. |
| [out] | TAU | TAU is DOUBLE PRECISION array of
dimension( max(NN) )
The Householder factors computed by DSYTRD in reducing A
to tridiagonal form. |
| [out] | Z | Z is DOUBLE PRECISION array of
dimension( LDU, max(NN) ).
The orthogonal matrix of eigenvectors computed by DSTEQR,
DPTEQR, and DSTEIN. |
| [out] | WORK | WORK is DOUBLE PRECISION array of
dimension( LWORK ) |
| [in] | LWORK | LWORK is INTEGER
The number of entries in WORK. This must be at least
1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2
where Nmax = max( NN(j), 2 ) and lg = log base 2. |
| [out] | IWORK | IWORK is INTEGER array,
Workspace. |
| [out] | LIWORK | LIWORK is INTEGER
The number of entries in IWORK. This must be at least
6 + 6*Nmax + 5 * Nmax * lg Nmax
where Nmax = max( NN(j), 2 ) and lg = log base 2. |
| [out] | RESULT | RESULT is DOUBLE PRECISION array, dimension (26)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow. |
| [out] | INFO | INFO is INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-5: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-23: LDU < 1 or LDU < NMAX.
-29: LWORK too small.
If DLATMR, SLATMS, DSYTRD, DORGTR, DSTEQR, SSTERF,
or DORMC2 returns an error code, the
absolute value of it is returned.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests performed, or which can
be performed so far, for the current matrix.
NTESTT The total number of tests performed so far.
NBLOCK Blocksize as returned by ENVIR.
NMAX Largest value in NN.
NMATS The number of matrices generated so far.
NERRS The number of tests which have exceeded THRESH
so far.
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTOVFL, RTUNFL Square roots of the previous 2 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) ) |
Definition at line 608 of file dchkst2stg.f.
1.9.5